Past Exam Questions

In this question all lengths are in centimetres.

The diagram shows triangle \(ABC\) with \(AB=\sqrt5-1\), \(AC=\sqrt5+1\) and \(BC=x\). The area of triangle \(ABC\) is \(\frac{2\sqrt5}{3}\text{ cm}^2\). Angle \(A\) is acute.

(a) Find the exact value of \(\sin A\).

(b) Find the exact value of \(\cos A\) and hence find the exact value of \(x\).

(c) Find the exact value of \(\sin B\).

The diagram shows triangle \(ABC\) with

\(AC=\sqrt6-\sqrt2,\quad AB=\sqrt6+\sqrt2\)

and angle \(CAB=60^\circ\).

(a) Find the exact length of \(BC\).

(b) Show that

\(\sin ACB=\frac{\sqrt6+\sqrt2}{4}.\)

(c) Show that the perpendicular distance from \(A\) to the line \(BC\) is \(1\).

The diagram shows a trapezium \(ABCD\), where \(AB\) is parallel to \(DC\), \(AD\) is perpendicular to \(DC\),

\(AB=2+3\sqrt5,\qquad DC=6+3\sqrt5,\qquad AD=10-2\sqrt5.\)

(i) Find the area of the trapezium in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

(ii) Find \(\operatorname{cot} BCD\) in the form \(c+d\sqrt5\), where \(c\) and \(d\) are constants.

Do not use a calculator in this question. In triangle \(ABC\), \(AB=2\sqrt5-1\), \(BC=2+\sqrt5\), and angle \(ABC=90^\circ\).

(i) Find the exact length of \(AC\).

(ii) Find \(\tan ACB\), giving your answer in the form \(p+q\sqrt r\), where \(p\), \(q\) and \(r\) are integers.

(iii) Hence find \(\operatorname{sec}^2ACB\), giving your answer in the form \(s+t\sqrt u\), where \(s\), \(t\) and \(u\) are integers.

The diagram shows the graph of \(y=3\cos2x-1\) for \(0^\circ\leqslant x\leqslant360^\circ\).

(a) Write down the amplitude and period of the graph.

(b) Sketch the graph, showing its key points.

In this question, all angles are in radians. (a) Write down the period of \(5 \tan \left(\frac{x}{4}\right)+1\).

(b) On the axes, sketch the graph of \(y=5 \tan \left(\frac{x}{4}\right)+1\) for \(-2 \pi \leqslant x \leqslant 4 \pi\).

State the intercept with the \(y\)-axis. Show clearly the positions of any asymptotes.

The diagram shows the curve \(y=a\cos bx+c\) for \(-180^\circ\leqslant x\leqslant180^\circ\).

It is given that \(a\), \(b\) and \(c\) are integers.

Find the values of \(a\), \(b\) and \(c\).

The diagram shows part of the graph of \(\mathrm{f}(x)=a \cos b x+c\), where \(a, b\) and \(c\) are constants. Given that \(\mathrm{f}(x)\) has a period of \(960^{\circ}\), find the values of \(a, b\) and \(c\).

The curve \(y=a \cos b x+c\), where \(a, b\) and \(c\) are integers, passes through the points \(\left(-\frac{\pi}{6},-2\right)\) and \(\left(\frac{\pi}{9}, \frac{1}{2}\right)\). The curve has a period of \(\frac{2 \pi}{3}\). (a) Find the values of \(a, b\) and \(c\). (b) Find the least value of \(y\) on the curve for \(0 \leqslant x \leqslant \frac{\pi}{2}\), and state the value of \(x\) at which this occurs.

On the axes, sketch the graph of \(y=4+5 \sin \frac{\theta}{2}\), for \(-360^{\circ} \leqslant \theta \leqslant 360^{\circ}\). State the intercept with the \(y\)-axis.

Given that \(y=2+4 \cos 3 \theta\), for \(-120^{\circ} \leqslant \theta \leqslant 120^{\circ}\), (a) write down the amplitude of \(y\)

(b) write down the period of \(y\).

(c) On the axes, sketch the graph of \(y\).

(a) The diagram shows the graph of \(y=a \cos b x+c\), for \(-360^{\circ} \leqslant x \leqslant 360^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

(b) The line \(y=p\) is a tangent to the curve \(y=3-2 \sin 6 \theta\). Write down the possible values of \(p\).

The diagram shows the graph of \(\quad y=a \sin b x+c\) for \(-360^{\circ} \leqslant x \leqslant 360^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

The diagram shows the graph of \(y=a \sin b x+c\), for \(-320^{\circ} \leqslant x \leqslant 320^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

The diagram shows the graph of \(y=a\cos bx+c\). Find the values of the constants \(a\), \(b\) and \(c\).

(a) Write down the period, in radians, of \(3\tan\frac{\theta}{2}-3\).

(b) On the axes, sketch the graph of \(y=3\tan\frac{\theta}{2}-3\) for \(-\pi\leq\theta\leq\pi\), stating the coordinates of the points where the graph meets the axes.

The function \(g\) is defined for \(0^\circ\leq x\leq120^\circ\) by

\(g(x)=2+4\cos6x.\)

(a) Sketch the graph of \(y=g(x)\).

(b) State the amplitude of \(g\).

(c) State the period of \(g\).

The diagram shows part of the graph of

\(y=a\cos\left(\frac{x}{b}\right)+c,\)

where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

The function \(g\) is defined by

\(g(x)=5\sin\left(\frac{3x}{4}\right)-2\)

for all values of \(x\).

(a) Write down the amplitude of \(g\).

(b) Write down the period of \(g\) in degrees.

(c) Sketch the graph of \(y=g(x)\), for \(-180^\circ\leq x\leq180^\circ\).

On the axes, draw the graph of

\(y=2\sin\frac{x}{3}-1\)

for \(-360^\circ \leq x \leq 360^\circ\).